ON THE INFLUENCE FUNCTIONS OF CERTAIN BIVARIATE MEDIANS

The asymptotic efficiency, influence function and breakdown properties of the multivariate median proposed by Oja are investigated and comparisons are made with two other generalizations of the median: the spatial median and the vector of marginal medians. It is shown that, for distributions with finite expectations, the Oja median has a bounded influence function although this median has 0% breakdown. For circular distributions, the influence functions of the Oja median and the spatial median coincide. The Oja median is then as efficient as the spatial median in the circular case but strictly better for any other elliptical case. It is noteworthy that the spatial median, however, has the best possible 50% breakdown. This phenomenon is analysed in the paper: it is shown that the finite sample breakdown point of the Oja median depends on the dispersion of the corrupted observations.